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Cartesian System
A plane formed by two number lines, one horizontal
and the other vertical, such that they intersect each
other at their zeroes, and then they form a Cartesian
Plane.
Scroll Down To Continue …
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Coordinate Axes:
The position of a point in a plane is fixed by selecting the axes of reference which are formed by two number lines intersecting each other at right angles, so that their zeroes coincide.
The horizontal number line is called x-axis and vertical number line is called y axis.
A point that lies on X Axis is (x,0)
A point that lies on Y Axis is (0,y)
Equation of Y Axis is x = 0
Equation of X Axis is y = 0
Equation of a lne parallel to Y Axis is x = a
Equation of a lne parallel to X Axis is y = a
Equation of a lne perpendicular to X Axis is X = a
Equation of a lne perpendicular to X Axis is X = a
The point of intersection of the two lines is called origin.
is the x-axis and Y1OY is the y-axis. These coordinate axes are also called rectangular axes as they are perpendicular to each other.
Rectangular coordinates are ordered pairs in which the first element is called the abscissa and the second element is called the ordinate.
● In the first quadrant, x is + ve and y is + ve
● In the second quadrant, x is – ve and y is + ve
● In the third quadrant, x is – ve and y is – ve
● In the fourth quadrant, x is + ve and y is -ve.
Distance Formula:
Example:
Example:
Collinearity of three points:
Three points P, Q and R are said to be collinear, if they lie in the same straight line.
i.e., PR = PQ + QR
i.e., PQ = PR + RQ
i.e., QR = QP + PR
If three points are not collinear, they always form a triangle.
Special Polygons:
(i) In Case of Triangle
(a) a right-angled triangle, if sum of squares of any two sides is equal to square of third side.
(b) an equilateral triangle, if all the three sides are equal.
(c) an isosceles triangle, if any two sides are equal.
(ii) In Case of Quadrilateral
(a) parallelogram, if opposite sides are equal and diagonals are not equal.
(b) rectangle, if opposite sides are equal and diagonals are equal.
(c) square, if all the four sides are equal and diagonals are equal.
(d) rhombus, if all the four sides are equal and diagonals are not equal.
Section Formula (Internal division only)
Midpoint Formula:
Point Dividing Two points in K : 1 Ratio:
Note:
If k is positive, the point divides the given points internally.
If k is Negative, the point divides the given points externally
Coordinates of the centroid of a triangle:
Points of Trisection:
If a line segment is divided into three equal parts by two points, the points are said to be the points of trisection.
In the given figure, the points R and S divide the line segment PQ into three equal parts i.e., PR=RS=SQ. The points R and S are said to be points of trisection.
Area of a Triangle:
The area of the triangle formed by the points
is calculated by the following expression.
Area of ∆PQR =
Area of Quadrilateral:
Area of a quadrilateral can be found by splitting up the quadrilateral into two triangles and sum up their areas.
Thus, area of quadrilateral PQRS = area of ∆PQR+ area of ∆PRS
Condition for collinearity of three points :
Three given points will be collinear, if the area of the triangle formed by these points is zero.
Rule to prove that three given points are collinear:
Step 1. Find the area of the triangle formed by the given points.
Step 2. Show that the area of the triangle formed by the given points is zero.
* The coordinates of the origin are O(0,0)
* The coordinates of any point on x-axis are (x, 0)
i.e., y=0 or ordinate is zero.
* The coordinates of any point on y – axis are (0, y) i.e., x=0 or abscissa is zero.
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An angle is positive if its rotation is in the anticlockwise and negative if its rotation is in the clockwise direction.
(Scroll down to continue …)
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If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ration of the angle can be determined.
Two angles are said to be complementary, if their sum is 900 and each one of them is called the complement of the other.
sin (900 – θ) = Cos θ
Cos (900– θ)= Sin θ
tan (900– θ) = Cot θ
Cot(900– θ) = tan θ
sec (900– θ)= cosec θ
cosec (900– θ) = sec θ
An equation with trigonometric ratios of an angle θ, which is true for all values of ‘ θ ‘, for which the given trigonometric ratios are defined, is called an identity.
The three fundamental trigonometric identities are
⇒ sin2 θ =1-cos2 θ
⇒ sin2 θ =(1-cos θ)(1+cos θ)
⇒ (1- cos θ) = (sin2 θ) /(1+ cos θ)
⇒ (1+ cos θ) = (sin2 θ) /(1- cos θ)
⇒ cos2 θ + sin2 θ = 1
cos2 θ =1- sin2 θ
⇒ cos2 θ =(1- sin θ)(1+ sin θ)
⇒ (1+ sin θ) = (cos2 θ) /(1- sin θ)
⇒ (1- sin θ) = cos2 θ /(1+sin θ)
(b) sec2 θ = 1 + tan2 θ
⇒ sec2 θ – tan2 θ =1
⇒ (sec θ – tan θ)(sec θ + tan θ) = 1
⇒ (sec θ – tan θ) = 1/ (sec θ + tan θ)
⇒ (sec θ + tan θ) = 1/ (sec θ – tan θ)
⇒ sec2 θ – 1 = tan2 θ
⇒ (sec θ – 1)( sec θ – 1) = tan2 θ
(c) cosec2 θ = 1+cot2θ
⇒ cosec2 θ – cot2 θ = 1
⇒ (cosec θ – cot θ)(cosec θ + cot θ)=1
(cosec θ+ cot θ) =1cosec θ – cot θ
(cosec θ- cot θ) = 1cosec θ + cot θ
⇒ Cosec2 θ – 1 = cot2 θ
⇒ (Cosec θ – 1)( Cosec θ – 1) = Cot2 θ
(a+b)2=+a2+b2+2ab
(a-b)2 = a2+b2-2ab
(a+b)2+(a-b)2= 2 (a2+b2)
(a+b)2– (a-b)2= 4ab
(a-b)2– (a+b)2= – 4ab
(a+b)2 = (a-b)2+ 4ab
(a-b)2 = (a+b)2– 4ab
(a2-b2)=(a+b)(a-b)
a+b=(a2-b2) /(a-b)
a-b=(a2-b2) /(a-b)
(a+b)2= (a-b)2+ 4ab
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Cartesian System
A plane formed by two number lines, one horizontal
and the other vertical, such that they intersect each
other at their zeroes, and then they form a Cartesian
Plane.
Scroll Down To Continue …
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Coordinate Axes:
The position of a point in a plane is fixed by selecting the axes of reference which are formed by two number lines intersecting each other at right angles, so that their zeroes coincide.
The horizontal number line is called x-axis and vertical number line is called y axis.
A point that lies on X Axis is (x,0)
A point that lies on Y Axis is (0,y)
Equation of Y Axis is x = 0
Equation of X Axis is y = 0
Equation of a lne parallel to Y Axis is x = a
Equation of a lne parallel to X Axis is y = a
Equation of a lne perpendicular to X Axis is X = a
Equation of a lne perpendicular to X Axis is X = a
The point of intersection of the two lines is called origin.
is the x-axis and Y1OY is the y-axis. These coordinate axes are also called rectangular axes as they are perpendicular to each other.
Rectangular coordinates are ordered pairs in which the first element is called the abscissa and the second element is called the ordinate.
● In the first quadrant, x is + ve and y is + ve
● In the second quadrant, x is – ve and y is + ve
● In the third quadrant, x is – ve and y is – ve
● In the fourth quadrant, x is + ve and y is -ve.
Distance Formula:
Example:
Example:
Collinearity of three points:
Three points P, Q and R are said to be collinear, if they lie in the same straight line.
i.e., PR = PQ + QR
i.e., PQ = PR + RQ
i.e., QR = QP + PR
If three points are not collinear, they always form a triangle.
Special Polygons:
(i) In Case of Triangle
(a) a right-angled triangle, if sum of squares of any two sides is equal to square of third side.
(b) an equilateral triangle, if all the three sides are equal.
(c) an isosceles triangle, if any two sides are equal.
(ii) In Case of Quadrilateral
(a) parallelogram, if opposite sides are equal and diagonals are not equal.
(b) rectangle, if opposite sides are equal and diagonals are equal.
(c) square, if all the four sides are equal and diagonals are equal.
(d) rhombus, if all the four sides are equal and diagonals are not equal.
Section Formula (Internal division only)
Midpoint Formula:
Point Dividing Two points in K : 1 Ratio:
Note:
If k is positive, the point divides the given points internally.
If k is Negative, the point divides the given points externally
Coordinates of the centroid of a triangle:
Points of Trisection:
If a line segment is divided into three equal parts by two points, the points are said to be the points of trisection.
In the given figure, the points R and S divide the line segment PQ into three equal parts i.e., PR=RS=SQ. The points R and S are said to be points of trisection.
Area of a Triangle:
The area of the triangle formed by the points
is calculated by the following expression.
Area of ∆PQR =
Area of Quadrilateral:
Area of a quadrilateral can be found by splitting up the quadrilateral into two triangles and sum up their areas.
Thus, area of quadrilateral PQRS = area of ∆PQR+ area of ∆PRS
Condition for collinearity of three points :
Three given points will be collinear, if the area of the triangle formed by these points is zero.
Rule to prove that three given points are collinear:
Step 1. Find the area of the triangle formed by the given points.
Step 2. Show that the area of the triangle formed by the given points is zero.
* The coordinates of the origin are O(0,0)
* The coordinates of any point on x-axis are (x, 0)
i.e., y=0 or ordinate is zero.
* The coordinates of any point on y – axis are (0, y) i.e., x=0 or abscissa is zero.
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Introduction :
One of the main applications of trigonometry is to find the distance between two or more than two places or to find the height of the object or the angle subtended by any object at a given point without actually measuring the distance or heights or angles.
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Trigonometry is useful to astronomers, navigators, architects and surveyors etc. in solving problems related to heights and distances.
The directions of the objects can be described by measuring :
(i) angle of elevation and (ii) angle of depression
Angles of elevation or angles of depression of the objects are measured by an instrument called The odolite.
The odolite is based on the principles of trigonometry, which is used for measuring angles with a rotating telescope.
In 1856, Sir George Everest first used the giant theodolite, which is now on display in the Museum of the survey of India in Dehradun.
Angle of Elevation:
Let P be the position of the object above the horizontal line OA and O be the eye of the observer, then angle AOP is called angle of elevation. It is called the angle of elevation, because the observer has to elevate (raise) his line of sight from the horizontal OA to see the object P. [ When the eye turns upwards above the horizontal line.]
Angle of Depression:
Let P be the position of the object below the horizontal line OA and O be the eye of the observer, then angle AOP is called angle of depression.
It is called the angle of depression because the observer has to depress (lower) his line of sight from the horizontal OA to see the object P.
[When the eye turns downwards below the horizontal line].
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Cartesian System
A plane formed by two number lines, one horizontal
and the other vertical, such that they intersect each
other at their zeroes, and then they form a Cartesian
Plane.
Scroll Down To Continue …
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Coordinate Axes:
The position of a point in a plane is fixed by selecting the axes of reference which are formed by two number lines intersecting each other at right angles, so that their zeroes coincide.
The horizontal number line is called x-axis and vertical number line is called y axis.
A point that lies on X Axis is (x,0)
A point that lies on Y Axis is (0,y)
Equation of Y Axis is x = 0
Equation of X Axis is y = 0
Equation of a lne parallel to Y Axis is x = a
Equation of a lne parallel to X Axis is y = a
Equation of a lne perpendicular to X Axis is X = a
Equation of a lne perpendicular to X Axis is X = a
The point of intersection of the two lines is called origin.
is the x-axis and Y1OY is the y-axis. These coordinate axes are also called rectangular axes as they are perpendicular to each other.
Rectangular coordinates are ordered pairs in which the first element is called the abscissa and the second element is called the ordinate.
● In the first quadrant, x is + ve and y is + ve
● In the second quadrant, x is – ve and y is + ve
● In the third quadrant, x is – ve and y is – ve
● In the fourth quadrant, x is + ve and y is -ve.
Distance Formula:
Example:
Example:
Collinearity of three points:
Three points P, Q and R are said to be collinear, if they lie in the same straight line.
i.e., PR = PQ + QR
i.e., PQ = PR + RQ
i.e., QR = QP + PR
If three points are not collinear, they always form a triangle.
Special Polygons:
(i) In Case of Triangle
(a) a right-angled triangle, if sum of squares of any two sides is equal to square of third side.
(b) an equilateral triangle, if all the three sides are equal.
(c) an isosceles triangle, if any two sides are equal.
(ii) In Case of Quadrilateral
(a) parallelogram, if opposite sides are equal and diagonals are not equal.
(b) rectangle, if opposite sides are equal and diagonals are equal.
(c) square, if all the four sides are equal and diagonals are equal.
(d) rhombus, if all the four sides are equal and diagonals are not equal.
Section Formula (Internal division only)
Midpoint Formula:
Point Dividing Two points in K : 1 Ratio:
Note:
If k is positive, the point divides the given points internally.
If k is Negative, the point divides the given points externally
Coordinates of the centroid of a triangle:
Points of Trisection:
If a line segment is divided into three equal parts by two points, the points are said to be the points of trisection.
In the given figure, the points R and S divide the line segment PQ into three equal parts i.e., PR=RS=SQ. The points R and S are said to be points of trisection.
Area of a Triangle:
The area of the triangle formed by the points
is calculated by the following expression.
Area of ∆PQR =
Area of Quadrilateral:
Area of a quadrilateral can be found by splitting up the quadrilateral into two triangles and sum up their areas.
Thus, area of quadrilateral PQRS = area of ∆PQR+ area of ∆PRS
Condition for collinearity of three points :
Three given points will be collinear, if the area of the triangle formed by these points is zero.
Rule to prove that three given points are collinear:
Step 1. Find the area of the triangle formed by the given points.
Step 2. Show that the area of the triangle formed by the given points is zero.
* The coordinates of the origin are O(0,0)
* The coordinates of any point on x-axis are (x, 0)
i.e., y=0 or ordinate is zero.
* The coordinates of any point on y – axis are (0, y) i.e., x=0 or abscissa is zero.
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Circle:
Circle is a round shaped figure has no corners or edges.
A circle is the locus of all points in a plane which are at constant distance
(called radius) from a fixed point (called centre). A circle with centre O and radius r is denoted by C (O, r).
Radius:
A line segement that joins the centre and circumference or boundary of the circles is called the radius of the circles.
A line segement that divides the circle into two halves is called he diameter of the circle.
Diameter = 2x radius
Radius = Diameter/2
Chord:
A line segment joining any two points on a circle. The largest chord of a circle is a diameter.
Position of A point With Respect To a Circle:
In a plane a point P can lie either inside, or on the circle or outside the given circle.
Position of A Line With Respect to A Circle
If a circle C(O, r) and a straight line ‘l’ are in the same plane, then only three possibilities are there. These are :
Outside The Circle:
(i) The line ‘l’ does not intersect the circle at all. The line ‘l’ is called a non-intersecting line with respect to the circle.
Inside of the Circle – Secant To A Circle:
The line ‘ l ‘ intersects the circle in two distinct points say A and B. The line which intersects the circle in two distinct points is called a secant line.
Touching The Circle – Tangent To Circle:
A tangent to a circle is a special case of the secant when the two end points of the corresponding chord are coincide.
That is the line ‘ l ‘ touches the circle in only one point. Such a line which touches the circle only in one point is called a tangent line.
Tangent To Circle :
Etimology of Tangent:
The word ‘tangent’ comes from the Latin word ‘tangere’, which means to touch and was introduced by the Danish mathematician Thomas Fincke in 1583.
Tangent is a line that intersects the circle in exactly one point.
A tangent to a circle is the limiting position of a secant when its two points of intersection with the circle coincide.
The common point of the circle and the tangent is called the point of contact.
In other words the point, at which the tangent touches the circle is called
the point of contact.
Number of Tangents From A Point To Circle:
Number of tangents to a circle from a point (say P) depends upon the position of the point P.
(a)
When point ‘P’ lies outside the circle: There are only two lines, which touch the circle in one point only, all the remaining lines either intersect in two points or do not intersect the, circle. Hence, there are only two tangents from point P to the circle.
(b)
When point ‘ P ‘ lies on the circle : There is only one line which touches the circle in one point, all other lines meet the circle in more than one point. Hence, there is one and only one tangent to the circle through the point P lies on the circle.
(c)
When point ‘ P ‘ lies inside the circle: Every line passing through the point P (lies inside the circle) intersect the circle in two points. Hence, there is no tangent through the point P lies inside the circle
There is only one tangent at a point on the circumference of the circle.
Point of contact is the common point of the tangent and the circle.
The tangent at any point of a circle is perpendicular to the radius through the point of
contact.
Theorems :
(i) Tangent-Radius Theorem
The line perpendicular to the tangent and passing through the point of contact, is known as the normal.
Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
The converse of above theorem is also true.
Theorem :
The tangents at any point of a circle is perpendicular to the radius through the point of contact. Or At the point of contact the angle between radius and tangents to a circle is 90^0 .
Theorem :
The length of tangents drawn from an external point to a circle are equal.
Important Results:
If two circles touch internally or externally, the point of contact lies on the straight line through the two centres.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
The length of the tangents drawn from an external point to a circle are equal.
PT =PT’ =
The distances between two parallel tangents drawn to a circle is equal to the diameter of the circle.
Facts:
In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
More Points To Remember !
There is no tangent to a circle passing through a point lying inside the circle.
At any point on the circle there can be one and only one tangent.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
There are exactly two tangents to a circle through a point outside the circle.
The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the tangent.
The lengths of the tangents drawn from an external point to a circle are equal.
The line containing the radius through the point of contact of tangent is called the normal to the circle at the point.
There is no tangent to the circle passing through a point lying inside the cirele.
There are exactly two tangents to a cirele through a point lying outside the circle
The length of the segment of the tangent from the external point and the point of contact
with the circle is called the length of the tangent.
The length of tangents drawn from an external point to a circle are equal.
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Introduction :
Circumference of a circle Or Perimeter of a circle :
The distance around the circle or the length of a circle is called its circumference or perimeter. (Scroll down to continue …)
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AREAS AND PERI METER RELATED TO CIRCLES
Introduction :
Circumference of a circle Or Perimeter of a circle :
The distance around the circle or the length of a circle is called its circumference or perimeter.
Circumference (perimeter) of a circle = d or 2r, where d is a diameter and r is a radius of the circle and = 227 or 3.14
Area of a circle:
Area of a circle = πr2
Area of a semicircle :
Area of semicircle = 12 πr2
Area of a quadrant :
Area of a quadrant (quarter circle) = r24
Perimeter of a semicircle:
Perimeter of a semicircle or protractor =πr+2r
Area of the ring :
Area of the ring or an annulus
= πR2 – πr2
= π(R2 – r2)
= π(R- r)(R+r)
Length of the arc AB
= 2πrθ3600 or πrθ1800
Area of a sector:
Area of sector OACBO =πrθ3600
OR
Area of sector OACBO = 12(r×l)
Perimeter of a sector :
Perimeter of sector OACBO
= Length of arc AB+2r
= πrθ1800+2r
Other important formulae :
(i) Distance moved by a wheel in 1 revolution = Circumference of the wheel.
(ii) Number of revolutions in one minute = Distance moved in 1 minuteCircumference
(iii) Angle described by minute hand in 60 minutes =3600
(iv) Angle described by hour hand in 12 hours = 3600
The midpoint of the hypotenuse of a right triangle is equidistant from the vertices of the triangle.
Angle subtended at the circumference by a diameter is always a right angle.
Area of a segment:
(i) Area of minor segment ACBA
= Area of sector OACBO- Area of ∆OAB
= πrθ3600- 12r2sinθ
(ii) Area of major segment BDAB
= Area of the circle – Area of minor segment ACBA
=πr2 – Area of minor segment ACBA
If a chord subtends a right angle at the center, then
Area of the corresponding segment = 4-12r2
If a chord subtends an angle of 600 at the center, then
Area of the corresponding segment = 6-32r2
If a chord subtends an angle of 1200 at the center, then Area of the corresponding segment = 3-34r2
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What is a cuboid?
The flat surfaces of a cuboid are known as its faces.
A cuboid has six faces, and each face is a rectangle.
These faces are arranged such that three pairs of opposite faces are parallel to each other.
The adjacent faces are perpendicular to each other (i.e., the angle between any two touching faces of a cube is right angle, 90°.
An edge is a line segment where the two surfaces of a cuboid meet.
There are 12 edges in a cuboid, where three edges meet at each vertex.
All edges form right angles with the adjacent edges and faces.
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cuboid has eight vertices.
Diagonal of a cuboid is a line segment that joins two opposite vertices.
The cuboid has four space diagonals.
Length of the diagonal of cuboid = √(length2 + breadth2 + height2) units.
Cuboids exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
It is a three-dimensional, Rectangular figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are rectangles.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cuboid are right angles.
The total surface area of a cuboid is defined as the area of its surface (Appearing face).
Imagine yourself sitting in a cuboid shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cuboid shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cuboid is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height)
Derivation of Total Surface Area of a Cuboid
Since the total surface area of a cuboid (TSA) is the area of its surface.
Total surface area of a cuboid = Lateral Surface Area + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Area Of Front Surface + Area Of Back Surface + Area Of Left Srface + Area Of Right Surface + Area Of Bottom Surface + Area Of Top Surface
Total surface area of a cuboid = Lateral Surface Area 2[Area Of Bottom Surface]
Since Area Of Top Surface = + Area Of Bottom Surface We get, Total surface area of a cuboid = Lateral Surface Area + 2[Area Of Top Surface]
TSA = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height) + (Length ✕ Breadth) + (Length ✕ Breadth)
TSA = 2(Length ✕ Height) + 2(Breadth ✕ Height) + 2(Length ✕ Breadth)
TSA = 2[(Length ✕ Height) + (Breadth ✕ Height) + (Length ✕ Breadth)]
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cuboid is defined as the space required for the cuboid or the Space occupied by the cuboid.
The volume of a cuboid can be calculated using the formula, V = lbh, where,
l = length, b = breadth or width, h = height
This formula shows that the volume of a cuboid is directly proportional to its length, breadth and height.
The volume is calculated by multiplying the object’s length, breadth, and height.
Hence the volume of the cube = lbh = lenth ✕ breadth ✕ height
Similar Shapes:
Real-world Examples:
Fun Fact:
Remember that these notes provide an overview of cuboids, and there are more advanced topics and applications related to this shape in various fields of study.
A cube is a three-dimensional regular polyhedron characterised by its 6 Identical (Congruent) Squares in which 3 Pairs of them parallel.
The flat surfaces of a cube are known as its faces.
A cube has six faces, and each face is a perfect square. These faces are arranged such that three pairs of faces are parallel to each other.
The adjacent faces are perpendicular to each other (the angle between any two touching faces of a cube is right angle, 90°.
All the edges have the same length.
A cube also has 8 vertices and 12 edges.
An edge is a line segment where the two surfaces of a cube meet.
There are twelve edges in a cube, where three edges meet at each vertex.
All edges have equal length and form right angles with the adjacent edges and faces.
A vertex is a point where the three edges meet. Vertices is the plural of vertex.
Cube has eight vertices.
The cube has four space diagonals that connect opposite vertices, each of which has a length of √3 times the length of an edge.
Cubes exhibit high symmetry.
They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.
It is a three-dimensional, square-shaped figure.
It has 6 faces, 12 edges, and 8 vertices.
All 6 faces are squares with equal area.
All sides have the same length.
Each vertex meets three faces and three edges.
The edges run parallel to those parallel to it.
All angles of a cube are right angles.
The total surface area of a cube is defined as the area of its outer surface.
Derivation of Total Surface Area of a Cube
Since the total surface area of a cube is the area of its outer surface.
total surface area of a cube = 6 ✕ area of one face.
We know that the cube has six square faces and each of the square faces is of the same size, the total surface area of a cube = 6 ✕ area of one face.
Let the length of each edge is “s”.
Area of one square face = length of edge ✕ length of edge
Area of one square face == s ✕ s = s²
Therefore, the total surface area of the cube = 6s²
The total surface area of the cube will be equal to the sum of all six faces of the cube.
Imagine yourself sitting in a cube shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.
That is, the lateral surface area of a cube shaped room is the area of its four walls, excluding the ceiling and the floor.
The lateral surface area of the cube is the sum of areas of its square faces, excluding the area of the top and the bottom face.
So the lateral surface area of a cube = sum of areas of 4 faces = 4a²
The Volume of a Cube
The volume of a three-dimensional object can be defined as the space required for it.
Similarly, Volume of a cube is defined as the space required for the cube or the Space occupied by the cube.
The volume of a cube can be calculated using the formula V = s3, where “s” represents the length of one side of the cube.
This formula shows that the volume of a cube is directly proportional to the cube of its side length.
The volume is calculated by multiplying the object’s length, breadth, and height. In the case of a cube shape, the length, width, and height are all of the same length. Let us refer to it as “s”.
Hence the volume of the cube is s ✕ s ✕ s = s³
We encounter many cubes in our daily life such as Ice cubes, sugar cubes, dice and the building blocks used in games.
Cubes play a fundamental role in the study of geometry and serve as a basis for understanding three-dimensional space and concepts such as volume and surface area.
Also, Cubes have many applications in mathematics, engineering, architecture and art etc.
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Cylinder is an important topic in Mathematics. It is a three-dimensional solid shape that has two parallel circular bases connected by a curved surface.
In this post, we will explore the properties of a cylinder and how to calculate its volume and surface area.
Let’s start with the basic definition of a right circular cylinder.
A cylinder is a solid shape that has two parallel circular bases of equal size and shape.
The curved surface that connects the two bases is called the lateral surface.
The axis of the cylinder is a line passing through the center of both bases.
(i). Solid Cylinder
(ii). Hollow Cylinder
Total Surface Area of Right Circular Cylinder = Curved Surface Area + Cicular Base Area + Circular Top Surface Area.
Since the Cicular Base Area And The Circular Top Surface Area of a cylinder are equal
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Top Surface Area)
Or
Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Base Surface Area)
Therefore the formula to calculate the surface area of a cylinder is expressed as the following:
$$SA = 2\pi r h + 2\pi r^2$$
where SA is the surface area, r is the radius of the base, and h is the height of the cylinder.
Here, Curved Surface Area, $$CSA = 2\pi r h$$ and
Cicular Base Area = Top Surface Area = $$\pi r^2$$
The formula to calculate the volume of a cylinder is given by:
$$V = \pi r^2 h$$
where V is the volume, r is the radius of the base, and h is the height of the cylinder.
Now let’s take an example to understand how to use these formulas. Suppose we have a cylinder with a radius of 4 cm and a height of 10 cm. To calculate its volume, we can use the formula:
$$V = \pi (4)^2 (10) = 160\pi$$
Therefore, the volume of the cylinder is 160π cubic cm.
To calculate its surface area, we can use the formula:
$$SA = 2\pi (4) (10) + 2\pi (4)^2 = 120\pi$$
Therefore, the surface area of the cylinder is 120π square cm.
In conclusion, understanding the properties of a cylinder and how to calculate its volume and surface area is important in CBSE Class 10 Mathematics. By using the formulas mentioned above, you can easily solve problems related to cylinders.
A hollow cylinder is a three-dimensional object with a circular base and a cylindrical shape. It is also known as a cylindrical shell. The cylinder has two circular faces and a curved surface. The thickness of the cylinder is uniform and it is hollow from inside.
The volume of a hollow cylinder can be calculated using the formula V = πh(R2-r2), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
The surface area of a hollow cylinder can be calculated using the formula A = 2πh(R+r), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.
Hollow cylinders are used in various applications such as pipes, drums, and containers. They are also used in engineering structures such as bridges and towers.
In conclusion, a hollow cylinder is a useful shape in various fields and can be easily calculated using mathematical equations.
A cone is a three-dimensional geometric shape that has a circular base and a single vertex. It can be visualized as a pyramid with a circular base. In this note, we will cover the basic concepts and formulas related to cones.
The surface area of a cone is the sum of the areas of its base and lateral surface. The formula to calculate the surface area of a cone is:
$$A = \pi r (r + l)$$
Where:
The volume of a cone is the amount of space enclosed by it. The formula to calculate the volume of a cone is:
$$V = \frac{1}{3} \pi r^2 h$$
Where:
Here are a few example equations related to cones:
Solid Sphere
Introduction – A solid sphere is a three-dimensional geometric figure in which all points inside the sphere are at the same distance from its center. – It is a type of 3D shape known as a “sphere” with a uniform density throughout.
Characteristics – The solid sphere has a well-defined volume, surface area, and mass. – It is completely filled with matter.
Volume of Solid Sphere The formula to calculate the volume (\(V_s\)) of a solid sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \] Where: \(V_s\) = Volume of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere.
Surface Area of Solid Sphere The formula to calculate the surface area (\(A_s\)) of a solid sphere is given by: \[ A_s = 4 \pi r^2 \]
Where: \(A_s\) = Surface area of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere
Mass of Solid Sphere The mass (\(m_s\)) of a solid sphere can be calculated using the formula: \[ m_s = \text{Density} \times V_s \] Where: \(m_s\) = Mass of the solid sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_s\) = Volume of the solid sphere (calculated using the previous formula)
Introduction – A hollow sphere is also a three-dimensional geometric figure, but unlike a solid sphere, it has an empty space inside. – It consists of an outer shell or surface with a certain thickness and an inner empty region.
Characteristics – The hollow sphere has a well-defined outer radius (\(R\)), inner radius (\(r\)), volume, surface area, and mass. – It is partially filled with matter, mainly in the form of the outer shell.
Volume of Hollow Sphere The formula to calculate the volume (\(V_h\)) of a hollow sphere is given by: \[ V_h = \frac{4}{3} \pi (R^3 – r^3) \] Where: – \(V_h\) = Volume of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Surface Area of Hollow Sphere The formula to calculate the surface area (\(A_h\)) of a hollow sphere is given by: \[ A_h = 4 \pi (R^2 – r^2) \] Where: \(A_h\) = Surface area of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere
Mass of Hollow Sphere The mass (\(m_h\)) of a hollow sphere can be calculated using the formula: \[ m_h = \text{Density} \times V_h \] Where: \(m_h\) = Mass of the hollow sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_h\) = Volume of the hollow sphere (calculated using the previous formula)
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Probability:
Probability is a measure of uncertainity (an event may happen or may not happen).
Probability is classified into the following two types.
(i) Empirical probability
(ii) theoretical probability.
Theoretical probability Or Classical Probability:
Theoretical probability Or Classical Probability is the probability of events based on the results obtained from theoretical approach.
Theoretical Approach
Here, we try to predict the outcomes without performing an actual experiment.
We assume that the outcomes of an experiment are equally likely.
We find that the experimental probability of an event approaches its theoretical probability if the number of trials of an experiment is very large.
Key Terms
Experiment:
Experiment is An activity that causes some well defined outcomes.
Random Experiment
Random experiment is an experiment that
may not give the same result on repitition
under identical conditions,
Trial: Each repition of the activity is called A Trial.
Outcome
Each possible result of a random experiment or a Trial is called its outcome.
Sample Space
The collection of all possible outcomes in a random experiment is called Sample space.
Or
The total number of possible outcomes of a random experiment.
As the number of trials in an experiment increases we may expect the empirical and theoretical probabilities to be nearly the same.
Event:
Any subset of smaple space is called an Event.
Compound Event An event connected to a random experiment is a compound event if it is obtained by combining two or more elementary events connected to the random experiment.
Occurrence of an event An event corresponding to a random experiment is said to occur if any one of the elementary events corresponding to the event is the outcome.
Impossible Events The event which never occurs is an impossible event.
So the probability ofan impossible eventis always zero.
Sure Event The event which certainly occurs is a sure event.In general, it is truethat for anevent E, ( ̅) ( ) Here theevent E is representing “not E”. This is called the compound ofthe event ‘E’.So ‘E’ and E are complementary events.
Theoretical Probability of An Event:
If there are n events associated with a random experiment and m of them are favorable to an event E, then the probability of the event E is denoted by P(E) and is calculated using the follwoing formula.
i.e.,
Where the event
representing non-occurrence of event E, is called the complement of the event E.
Hence, E and
are complementary events.
Sum of the probabilities of all the elementary events of an experiment is
i.e.,
P (E1) + P (E2) + P (E3) … + P (En) = 1
Playing Cards:
A pack of cards consists of four suits. They are
Each suit consists of 13 cards, nine cards numbered 2, 3, 4 ….10, an Ace (B) a Jack (J), a Queen (Q), and a King (K).
Spades and Clubs are black in colour.
Hearts and Diamonds are red in colour.
So there are 26 black cards and 26 red cards.
King, queen and jack are called face cards.
There are totally 12 (4 x3) face cardsin a packof 52 cards.
I.e. in eachsuit we have3 face cards.
Coins:
A coin has two sides namely head and tail.
In the experiment of tossing a coin for once, there are 2 possible outcomes. They are 1 head, 1 tail.
P (Head) = 1/2 = P (Tail)
Dice:
A die is a well-balanced cubewith six facesnumbered from 1 to 6.
Dice is the pluralform. There are six equally likely outcomes 1, 2,3, 4, 5,6 in a single throw. Geometric Probability:
If the total number of outcomes of a trial in a random experiment is infinite, then the above definition is not sufficient to find the probability of an event.
In such cases, the definition of probability is modified and probability so obtained is called Geometric Probability.
The geometric probability p of an event is given by
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Nutrients: Food substances that provide nourishment to the body are called nutrients.
The major nutrients in our food are carbohydrates, proteins, fats, vitamins and minerals. In addition, food also contains dietary fibres and water. Carbohydrates and fats mainly provide energy to our body. (Scroll down to continue …)
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Carbohydrates: Carbohydrates are energy-giving compounds.
Cellulose, starch and sugar are the examples of carbohydrates.
There may be simple carbohydrates or complex carbohydrates.
Fats: Fats are very high energy-giving compounds. They produce greater amount of energy than carbohydrates.
Sources of fats: animal fats and vegetable fats.
Carbohydrates and fats are Energy giving food.
Minerals: minerals are the elements required by the body in small amounts. Minerals are essential for growth and development of bones, teeth and red blood cells.
Proteins: These are body-building foods. They help in growth of the body.
Vitamins: These are organic substances that protect the body from diseases.
Roughage: It is the dietary fibre present in the food. It facilitates regular movement of the bowels and prevents constipation.
Note: Dietary fibre and water are not food.
Balanced diet: It provides all the nutrients that our body needs, in right quantities, along with adequate amount of roughage and water.
Deficiency Diseases: These are the diseases cause due to the lack of required nutrients for a long period in the diet.
Malnutrition: when a person eats enough of food but his diet is unbalanced, it isknown as malnutrition.
Undernutrition: A person not eating sufficient food to maintain good health suffers from undernutrition.
Some Nutrients Deficiency Diseases:
Protein: Kwashiorkar – Stunted growth, thinning of legs, protruding belly.
Protein and Carbohydrates – Marasmus – Complete/partial arrest of growth, lack of energy.
Vitamin D and calcium: Rickets – Bowed legs, bent spine, deformed bones are joints.
Vitamin C: Scurvy – Bleeding and swelling of gums, weakness.
Iodine: Goitre – Enlargement of thyroid gland, retarded growth.
Iron – Anaemia – Fatigue, loss of appetite, pale skin.
Vitamin K: Bleeding disease- delay in blood clotting leads to excess bleeding.
Beri-beri: Vitamin B1 – weakness in muscles, little energy to do work, paralysis
Night blindness – Vitamin A – No vision at night or in dim light.
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Material: A material is a substance which is used for making things.
A given material could be used to make a large number of objects.
It is also possible that an object could be made of a single material or of many different types of materials. (Scroll down to continue …)
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GENERAL PRPERTIES OF MATRIAL
(A) Apearance : Shiny ( luster or luster ) or Roughness ( non-lustrous ) All metals are lustrous , some are more and some are less.
wood, rubber or a piece of rock is non-lustrous.
(B) Hardness: Hard and soft Rocks, iron and many metals are hard.
Hard materials may be:
1. Brittle :- Which break into samller pieces or are powdered when hammered.
Example: rock, glass, salt.
Malleable: Which spread into sheets when beaten. metals are malleable.
Ductile:- they can be drawn into thin and long wires. metals are ductile.
Materials like wood, rubber and fibres are soft.
(C) Visibility:-
Transarent :- object through which light passes .
Examples: glass some plastis, sugar and salt solution in water etc.
Transluscent:- object through which light passes partially.
Example: Butter paper, etc.
Opaque: object through which light does not pass.
Examples: wood, metals, rock and wall etc.
(D) Good and bad conductor of electricity:-
Good conductors of electricity:- Material through which electric current can pass easily are the good conductors of electricity.
Examples: metals are good conductors of electricity.
Bad conductors of electricity:- Material through which electric current do not can pass are the bad conductors of electricity.
Examples: wood, plastic, rubber, cork and certain materials are bad conductors of electricity.
(D) Good and bad conductor of heat:- All metals are good conductors of heat.
Examples: wood are bad conductors of heat.
(E) combustible substances:- The materials which catch fire on heating are combustible substance.
wood, plastic, fibre and paper are combustible substance.
(F) SOME MATERIAL MAY FLOAT AND SOME SINK IN WATER.
material which sinks in water is denser than water. material which floats on water is less dense than water.
(G) SOLUBILITY O A SUBSTANCES IN WATER.
The substnces like salt and sugar , which disappear in water , are said to be soluble.
The substances which remain as such and do not disappear in water are said to be
(H) Miscible and Immiscible Liquid:- when two liquids are mixed and they do nort mixed well, they are said to be immiscible.
The liquids which mix well with water are said to be miscible.
GROUPING OF MATERIALS. Things made from wood
Things made from metal
Things made from ceramic
Things made from glass
Things made from plastic
Things made from rubber
things that are liquids such as oil.
Materials are grouped together on the basis of similarities and differences in their properties.
Things are grouped together for convenience and to study their properties.
Advantages of classification:
(a) Helps in identification of objects.
(b) Helps in sorting of objects.
(c) Helps in locting things.
(d) Makes study of different objects easy and more meaningful rather than studying each other separately. (e) Helps to understanding similarities and dissimilarties among
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PURE SUBSTANCES: The substances which contain only one kind of particles are called pure substances. Elements and compounds are pure substances.
ELEMENT: A substance made from identical particles of one material.
COMPOUND: A substance formed as a result of chemical combination of two or more elements in a fixed ratio.
Solution: A solution is a homogeneous mixture of two substances.
Solvent: The substance in larger quantity is the solvent.
The substance other than the solvent present in the solution is/are is the solute/solutes.
IMPURE SUBSTANCES: The substances which contain more than one kind of particles are called impure substances.
NEED FOR SEPARATING COMPONENT OF A MIXTURE
Removing harmful or unwanted components, and obtaining useful and desire component in pure form.
Methods of separation are Threshing, winnowing, handpicking, sieving, magnetic separation, floating and sinking method, filtration.
separation to immiscible liquids, churning to separate cream from milk, sublimation.
(i) Separation of solid from other solids:
(a) Threshing: The process of separating grain from husk or chaff is called threshing.
Chaff: The waste material of agriculture processes such as pieces of straw are called chaff.
(b) Winnowing: The process of separation of heavier and lighter components of a mixture by wind or blowing air.
(c) Hand-picking: The process of picking the undesirable components from desirable material with hand is called Hand-picking.
(d) Sieving: The process of separating the solid constituents of different sizes in a mixture using a sieve is called sieving.
Sieve: A sieve is a device with many small holes in it , which allow the smaller particles to pass through and stops the bigger particles in it is called a sieve.
(e) Magnetic separation: Process of seperating the magnetic material by moving a magnet over a mixture of magnetic and non-magnetic substances is called magnetic seperation. In this process the magnetic material sticks to the magnet are removed.
(ii) Separation of water soluble solids or separating solute dissolved in solvent:
(a) Evaporation: Process of converting a liquid into its gaseous state by placing in open air without heating is called evaporation.
(b) Condensation: Process of converting a gaseous material into its liquid state by cooling it is called condensation.
(iii) Separation of insoluble solids from Liquids:
(a) Sedimentation: The process of settling down of heavier and insoluble component from mixture is called sedimentation.
Examples: sand, water.
(b) Decantation: The process of transferring clean liquid without disturbing the sediment, is called decantation
(c) Loading: The methods by which finer particles are made to settle faster by dissolving a small quantity of alum.
(d) Filtration: In this process the impurities are passed through a filter.
Filter: The filter has pores in it that allow only liquids to pass through the pores but stops the suspended particles or solid particles.
Therefore, the filter separates the suspended particles or solid particles.
The clear liquid so obtained in filtration is called the filtrate.
residue: The left over material on the filter paper due to filtration is known as residue.
(iv) Separation of Immiscible Liquids:
IMMISCIBLE: The liquids which donot mix well with one another are called immiscible liquids.
Example: Mixture of oil and water are immiscible liquids.
immiscible liquids are sepreated by centrifugation, churning etc. methods.
(b) Centrifugation: The method in which mixture containing suspended particles is rotated at a high speed in a centrifuge and heavier particles settle down.
Example: Cream is separated from milk, using centrifugation method.
(c) Churning: The method which is used for separating lighter particles of solid suspended in.
SEPARATING SOLUTE NOT SOLUBLE IN A SOLVENT
Flotation and sinking methods:- This method is used when the components are not soluble in water and one of the component of a mixture is lighter than water and the other is heavier than water.
SUBLIMATION:- The changing of a solid directly to vapour, without coming to liquid state is called sublimation.
Husk and stones could be separated from grains by handpicking.
· Husk is separated from heavier seeds of grain by winnowing.
· A saturated solution is a solution in which no more solute could be dissolved.
· More of a substance can be dissolved in a solution by heating it.
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Plants are usually grouped into herbs, shrubs, trees, creepers and climbers.
From where does a plant come?
A plant comes from a seed. A seed is covered with seed coats.,The seed coats protect the seed. The seed has a baby plant inside. (Scroll down to continue …)
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The baby plant has a tiny root towards the outer side. The tiny root is termed as radicle.
The baby plant has a tiny shoot towards the inner side. The tiny shoot is termed as plumule.
Classification of Plants Based On The of Growth Habit:
(a) Herbs: Have soft, green and weak stems.
Example: rice, wheat, maize, sunflower, mint,etc.
(b) Shrubs: They are bushy and have hard stems that do not bend easily.
These are plants with the stem branching out near the base. Example: lemon, China rose, jasmine, Nerium, etc.
(c) Trees: These are big plants which have a tall and strong stem (trunk).
Stems have branches in the upper part, much above the ground. Live for many years. Example: mango,neem, banyan, coconut, etc.
(d) Climbers: Have weak stems and cannot stand erect. They take the support of other treesand climb on them. Example: pea, grape, vine, etc.
(e) Creepers: Plants which creep on the ground and spread out. Example: pumpkin and watermelon.
Classification of plants on the basis of their Life Cycle:
(a) Annuals: Plants whose life cycle is completed in the one season. These are generallyherbs.
Example: wheat and mustard.
(b) Biennials: Plants whose life cycle requires two seasons for completion. They aregenerally
herbs and rarely shrubs.
Example: carrot, radish, and potato.
(c) Perennials: Plants whose life cycle runs for more than two seasons example: guava, Babul, and palm trees. Parts of a Plants:
(a) Root system: The root and its branches make up the root system of a plant. Root is the underground art of a plant body.It is non-green. The root grows into the soil and away from the light.
(i) Tap Root: It is formed by the baby root (radicle) which comes out from a germinating seed.
It is the main primary root arises from the lower end of the stem.
A number of tiny branches called secondary roots. Example: mustard, neem, rose, etc.
(ii) Fibrous Root: A bunch of roots arises from the base of the stem. Example: wheat, maize, etc. Functions of root system:
(i) Roots absorb water and nutrients from the soil.
(ii) Roots help the plant to stand erect.
(iii) Roots check soil erosion.
(iv) Roots store food.
(v) Prop roots offer extra support.
MODIFIED ROOTS:
Roots may show certain changes in their shape, size or structure for performing special functions. These roots are called the modified root.
1. Roots modified for the storage of food: – We find such roots in carrots, radish, turnip, beetroot, sweet potato and some other plants.
2. Roots modified to provide support to the plant: – In a banyan tree, you might have noted root like roots coming out from the branches and hanging downwards.
These roots give support to the branches. They are called the supporting roots or the prop roots.
3. Root modified to bear nodules: On the roots of the gram and other pulses.
Rhizobium, a nitrogen-fixing bacteria makes its home in the form of small are termed as nodules.
(b) Shoot system: The part of the plant which grows above the soil.
It includes The stem, The leaves,The buds, The branches,The flower, The fruits, The seeds.
(ii) Leaf: A leaf is a flat and green structure on a plant, coming out from a node on the stemand always bearing a bud in its axil. Parts of leaf:
(a) Petiole: Stalk of the leaf with which it joined to the stem.
(b) Leaf lamina: The flat green portion of the leaf. (c) Veins: These are the network of small, narrow, tube-like structures on both sides The function of Leaf:
(i) Transpiration: Process of losing water by the leaves of a plant.
(ii) Preparation of food by the process of photosynthesis.
(iii) Flower: It is the reproductive organ of the plant.
The function of Flowers:
(i) Help in reproduction.
(ii) These become fruits that store food and seeds.
(iii) Modified flowers like cauliflower, broccoli are rich sources of vitamins. ·
Parts of Flower:
(i) Pedicel: Stalk of the flower through which the flower is joined to the branch. It has joinedto the branch.
(ii) Sepal: Small green leaf-like structures on the thalamus. They protect the flower.
(iii) Petals: Brightly colored leaf-like structures present inside the sepals. Petals attract the insects and help in the process of reproduction.
(iv) Stamens: These are long, thin and needle-like structures. These are male organs of the flower. It consists of two parts: Anther, Filament.
(a) Anther: The swollen tip of each stamen that encloses in it a small powdery
(b) Filament: Long stalk-like structure that joins the anther with thalamus.
(v) Carpel: It is a flask-shaped organ in the center of the flower. It is the female organ of the flower. It consists three parts: Style, Stigma, and Ovary.
(a) Style: a Long thin tube-like structure which is swollen at the base.
(b) Stigma: Small, round and sticky part of the carpel at the top of the style the
traps thepollen grains.
(c) Ovary: Swollen part of carpel that contains ovules. The Bud : A bud is a compact or a condensed shoot. It encloses future stem, nodes and leaves. Axillary bud Terminal bud floral bud
(i) Stem: Stem forms the central axis of the plant body.Gives rise to a number of branches that bear leaves. The stem bears leaves, flowers, and fruits.
The function of Stem:
(i) It provides support to plant.
(ii) It bears important plant parts.
(iii) It helps in transportation of water and food.
(iv) Underground stems store food.
(v) Thick and fleshy stems make food.
(vi) Stem modified into tendrils give extra support for Some Extra Points:
The pattern of veins on the leaf is called venation. It can be reticulate or parallel.
Leaves give out water vapor through the process of transpiration.
Green leaves make their food by the process of photosynthesis using carbon dioxide and water in the presence of sunlight.
Roots absorb water and minerals from the soil and anchor the plant firmly in the soil.
Roots are mainly of two types: taproot and fibrous roots.
Plants having leaves with reticulate venation have tap roots while plants having leaves with parallel venation have fibrous roots.
The stem conducts water from roots to the leaves (and other parts) and food from leaves to other parts of the plant.
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